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One kind of simple crystal radio consists of the series LRC circuit drawn above. The antenna-to-ground connection represents an amplitude-modulated AC voltage source $V = V(t)\cos(\omega t)$ where $\omega = 2\pi \times 10^6$ radians/sec. The diode (in series with the earphone of resistance $R$) is a circuit element that only lets current flow in the direction of the arrow - a one-way gate. The capacitor can be varied to tune the radio.

a) How should the capacitor be set (with respect to the value of $L$) to tune the radio to deliver the maximum current through the earphones (resistance $R$) and what is that maximum current? Assume that the diode has negligible resistance and capacitance and that the amplitude-modulation is slow relative to $\omega^{-1}$.

b) Describe qualitatively, with a suitable picture or figure, how the diode allows the amplitude-modulated signal ($V(t)$) to be extracted from the carrier frequency $\omega$.